Expanding cone and applications to homogeneous dynamics

Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results of translation of $U$-slice by elements of $A_U^+$ on finite volume homogeneous space $G/\Gamma$ where $H$ is a subgroup of $G$. More precisely, we prove quantitative nonescape of mass and equidistribution of $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma$ has a spectral gap we prove effective $k$-equidistribution of $U$-slice and pointwise equidistribution for one parameter semi-flow in $A_U^+$ with an error rate. These results are only known before in special cases such as the space of unimodular lattices in Euclidean space. In the paper we formulate expanding cone and dynamical results above in more general setting where $H$ is a semisimple Lie group without compact factors.